Finite Group Databasebeta

← #189961

Subgroups — #189961

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2#21847C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4#8727C212yesmaximal, complemented
5#8727C212yesmaximal, complemented
6#8725C212yesmaximal, complemented
7#8724C212yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#8043C211yesmaximal, characteristic, complemented
10#8039C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#520—21no—
13C2×C4×#41C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×#408—21no—
19Q8×C4⋊C4—21no—
20Q8×C4⋊C4C412yes—
21D4×C4⋊C4—22no—
22D4×C4⋊C4—21no—
23D4×C4⋊C4C412yes—
24D4×C4⋊C4C414yes—
25#2317—22no—
26#2317—21no—
27#2285C2212yes—
28#2285C2211yescharacteristic
29C2×#384C2212yes—
30C2×#384C2211yescharacteristic
31#1984—21no—
32#1954—21no—
33#1869—22no—
34#1798—22no—
35#1797C2212yes—
36#1797C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1729—21no—
40#1296C2211yescharacteristic
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47#1296C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2212yes—
50C22×#41D411yescharacteristic
51C22×M16—21no—
52C23×C8—21no—
53C23×C8—21no—
54#520—44no—
55#520—44no—
56C4×#41—22no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—24no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×#117—22no—
70C2×#117—21no—
71C2×#117D412yes—
72C2×#117D411yescharacteristic
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C414yes—
78C22×C4⋊C4C2×C412yes—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83#423—22no—
84#423—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98#408—44no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C4×C8—21no—
105#389—22no—
106#389—24no—
107#389—22no—
108#389—24no—
109#385—24no—
110#385—24no—
111#385—24no—
112#384—22no—
113#384—21no—
114#384—22no—
115#384—21no—
116#384C2×C414yes—
117#384C2×C412yes—
118#384C2311yescharacteristic
119#384C2312yes—
120#384C2311yescharacteristic
121#382—22no—
122#382—21no—
123#380—22no—
124#380—21no—
125#380C2×C414yes—
126#380C2×C412yes—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C414yes—
130C4×C4⋊C4C2×C412yes—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—22no—
139C2×#41SD1614yes—
140C2×#41Q1614yes—
141C22×Q8C22⋊C411yescharacteristic
142C22×D4C22⋊C412yes—
143C22×D4C22⋊C411yescharacteristic
144C23×C4—24no—
145C23×C4—22no—
146C23×C4C22×C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C2×M16—42no—
153C2×M16—42no—
154C2×M16—42no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—42no—
162C22×C8—42no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8—81no—
166C22×C8—81no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—48no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—416no—
181C4×D4—48no—
182#117—22no—
183#117—22no—
184#117—24no—
185#117—24no—
186#117—44no—
187#117—44no—
188#117—42no—
189#117—42no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—84no—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C22×C412yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22⋊C412yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42C22×C412yes—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42C2411yescharacteristic
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C22⋊C8—42no—
257C22⋊C8—44no—
258C22⋊C8—44no—
259C22⋊C8—82no—
260C22⋊C8—82no—
261C4×C8—42no—
262#95—24no—
263#95—22no—
264#95—24no—
265#95—22no—
266C24C2×C22⋊C412yes—
267C24C2×C22⋊C411yescharacteristic
268#41—28no—
269#41—28no—
270#41—28no—
271#41—28no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8#10314yes—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4#10318yes—
280C2×D4#10314yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—48no—
305C22×C4—44no—
306C22×C4—82no—
307C22×C4—84no—
308C22×C4—82no—
309C22×C4—84no—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4#11712yes—
317C22×C4#11712yes—
318C22×C4#12312yes—
319C22×C4#12312yes—
320C22×C4#12312yes—
321C22×C4#12312yes—
322C22×C4C2×SD1614yes—
323C22×C4C2×SD1612yes—
324C22×C4C2×Q1614yes—
325C22×C4C2×Q1612yes—
326C22×C4AGL(1,ℤ/8)12yes—
327C22×C4AGL(1,ℤ/8)12yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331M16—84no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—84no—
351C2×C8—84no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—48no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—44no—
373C4⋊C4—82no—
374C4⋊C4—84no—
375C4⋊C4—82no—
376C4⋊C4—84no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—48no—
381C22⋊C4—44no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—48no—
389C42—44no—
390C42#11712yes—
391C42#11712yes—
392C42#12312yes—
393C42#12312yes—
394C42#12312yes—
395C42#12312yes—
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23#38112yes—
411C23C2×#10314yes—
412C23C2×#10312yes—
413C23#41812yes—
414C23C22×C22⋊C411yescharacteristic
415Q8—28no—
416D4—216no—
417D4—28no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—164no—
458C2×C4—168no—
459C2×C4#38112yes—
460C2×C4#38112yes—
461C2×C4#38112yes—
462C2×C4C2×#10314yes—
463C2×C4C2×#10312yes—
464C2×C4#41812yes—
465C2×C4#48114yes—
466C2×C4#48114yes—
467C2×C4#48414yes—
468C2×C4#48414yes—
469C2×C4C2×#11711yescharacteristic
470C2×C4C2×#11711yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4C2×#12311yescharacteristic
474C2×C4C2×#12311yescharacteristic
475C2×C4C22×SD1611yescharacteristic
476C2×C4C22×Q1611yescharacteristic
477C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
478C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
479C2×C4#57611yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57811yescharacteristic
483C8—81no—
484C8—81no—
485C8—82no—
486C8—82no—
487C8—81no—
488C8—81no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—24no—
496C22—24no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22#182414yes—
503C22C2×#38111yescharacteristic
504C22#288811yescharacteristic
505C22#288911yescharacteristic
506C22C22×#10311yescharacteristic
507C22C2×#41811yescharacteristic
508C22#289711yescharacteristic
509C22#289811yescharacteristic
510C4—21no—
511C4—21no—
512C4—22no—
513C4—24no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—84no—
518C4—82no—
519C4—82no—
520C4#182414yes—
521C4C2×#48111yescharacteristic
522C4C2×#48111yescharacteristic
523C4C2×#48411yescharacteristic
524C4C2×#48411yescharacteristic
525C4#309611yescharacteristic
526C4#309711yescharacteristic
527C4#309811yescharacteristic
528C4#309911yescharacteristic
529C2—24no—
530C2—44no—
531C2#1772211yescharacteristic
532C2#1772311yescharacteristic
533C2#1775811yescharacteristic
534C2#1775911yescharacteristic
535C2#2069511yescharacteristic
536C2C2×#182411yescharacteristic
537C2#2070311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #8727C2semidirect
5. #8727C2semidirect
6. #8725C2semidirect
7. #8724C2semidirect
8. #8045C2semidirect
9. #8043C2semidirect
10. #8039C2semidirect
11. C2×#1296C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#341832
#218472
#206292
#87272
#87252
#87242
#80452
#80432
#80392
C2×#12962